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Group Sequential Design for One-Sample Slope

Source: R/getDesignMeans.R
getDesignOneSlope.Rd

Obtains the power given sample size or obtains the sample size given power for a group sequential design for one-sample slope.

Usage

getDesignOneSlope(
  beta = NA_real_,
  n = NA_real_,
  slopeH0 = 0,
  slope = 0.5,
  stDev = 1,
  stDevCovariate = 1,
  normalApproximation = TRUE,
  rounding = TRUE,
  kMax = 1L,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  futilityStopping = NA_integer_,
  criticalValues = NA_real_,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  futilityBounds = NA_real_,
  typeBetaSpending = "none",
  parameterBetaSpending = NA_real_,
  userBetaSpending = NA_real_,
  spendingTime = NA_real_
)

Arguments

beta

The type II error.

n

The total sample size.

slopeH0

The slope under the null hypothesis. Defaults to 0.

slope

The slope under the alternative hypothesis.

stDev

The standard deviation of the residual.

stDevCovariate

The standard deviation of the covariate.

normalApproximation

The type of computation of the p-values. If TRUE, the variance is assumed to be known, otherwise the calculations are performed with the t distribution. The exact calculation using the t distribution is only implemented for the fixed design.

rounding

Whether to round up sample size. Defaults to 1 for sample size rounding.

kMax

The maximum number of stages.

informationRates

The information rates. Fixed prior to the trial. Defaults to (1:kMax) / kMax if left unspecified.

efficacyStopping

Indicators of whether efficacy stopping is allowed at each stage. Defaults to TRUE if left unspecified.

futilityStopping

Indicators of whether futility stopping is allowed at each stage. Defaults to TRUE if left unspecified.

criticalValues

Upper boundaries on the z-test statistic scale for stopping for efficacy.

alpha

The significance level. Defaults to 0.025.

typeAlphaSpending

The type of alpha spending. One of the following: "OF" for O'Brien-Fleming boundaries, "P" for Pocock boundaries, "WT" for Wang & Tsiatis boundaries, "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early efficacy stopping. Defaults to "sfOF".

parameterAlphaSpending

The parameter value for the alpha spending. Corresponds to \(\Delta\) for "WT", \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userAlphaSpending

The user defined alpha spending. Cumulative alpha spent up to each stage.

futilityBounds

Lower boundaries on the z-test statistic scale for stopping for futility at stages 1, ..., kMax-1. Defaults to rep(-6, kMax-1) if left unspecified. The futility bounds are non-binding for the calculation of critical values.

typeBetaSpending

The type of beta spending. One of the following: "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early futility stopping. Defaults to "none".

parameterBetaSpending

The parameter value for the beta spending. Corresponds to \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userBetaSpending

The user defined beta spending. Cumulative beta spent up to each stage.

spendingTime

A vector of length kMax for the error spending time at each analysis. Defaults to missing, in which case, it is the same as informationRates.

Value

An S3 class designOneSlope object with three components:

  • overallResults: A data frame containing the following variables:

    • overallReject: The overall rejection probability.

    • alpha: The overall significance level.

    • attainedAlpha: The attained significance level, which is different from the overall significance level in the presence of futility stopping.

    • kMax: The number of stages.

    • theta: The parameter value.

    • information: The maximum information.

    • expectedInformationH1: The expected information under H1.

    • expectedInformationH0: The expected information under H0.

    • drift: The drift parameter, equal to theta*sqrt(information).

    • inflationFactor: The inflation factor (relative to the fixed design).

    • numberOfSubjects: The maximum number of subjects.

    • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

    • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

    • slopeH0: The slope under the null hypothesis.

    • slope: The slope under the alternative hypothesis.

    • stDev: The standard deviation of the residual.

    • stDevCovariate: The standard deviation of the covariate.

  • byStageResults: A data frame containing the following variables:

    • informationRates: The information rates.

    • efficacyBounds: The efficacy boundaries on the Z-scale.

    • futilityBounds: The futility boundaries on the Z-scale.

    • rejectPerStage: The probability for efficacy stopping.

    • futilityPerStage: The probability for futility stopping.

    • cumulativeRejection: The cumulative probability for efficacy stopping.

    • cumulativeFutility: The cumulative probability for futility stopping.

    • cumulativeAlphaSpent: The cumulative alpha spent.

    • efficacyP: The efficacy boundaries on the p-value scale.

    • futilityP: The futility boundaries on the p-value scale.

    • information: The cumulative information.

    • efficacyStopping: Whether to allow efficacy stopping.

    • futilityStopping: Whether to allow futility stopping.

    • rejectPerStageH0: The probability for efficacy stopping under H0.

    • futilityPerStageH0: The probability for futility stopping under H0.

    • cumulativeRejectionH0: The cumulative probability for efficacy stopping under H0.

    • cumulativeFutilityH0: The cumulative probability for futility stopping under H0.

    • efficacySlope: The efficacy boundaries on the slope scale.

    • futilitySlope: The futility boundaries on the slope scale.

    • numberOfSubjects: The number of subjects.

  • settings: A list containing the following input parameters:

    • typeAlphaSpending: The type of alpha spending.

    • parameterAlphaSpending: The parameter value for alpha spending.

    • userAlphaSpending: The user defined alpha spending.

    • typeBetaSpending: The type of beta spending.

    • parameterBetaSpending: The parameter value for beta spending.

    • userBetaSpending: The user defined beta spending.

    • spendingTime: The error spending time at each analysis.

    • normalApproximation: The type of computation of the p-values. If TRUE, the variance is assumed to be known, otherwise the calculations are performed with the t distribution.

    • rounding: Whether to round up sample size.

Details

We assume a simple linear regression of the form $$y_i = \alpha + \beta x_i + \epsilon_i$$ where \(\epsilon_i\) is the residual error, which is assumed to be normally distributed with mean 0 and standard deviation \(\sigma_\epsilon\). The covariate \(x_i\) is assumed to be normally distributed with mean 0 and standard deviation \(\sigma_x\). The slope under the null hypothesis is \(\beta_0\), and the slope under the alternative hypothesis is \(\beta\). Since $$\hat{\beta} = \frac{\sum_{i=1}^{n} (x_i-\bar{x}) y_i} {\sum_{i=1}^{n}(x_i-\bar{x})^2}$$ it follows that $$\hat{\beta} \sim N(\beta, \frac{\sigma_\epsilon^2}{\sum_{i=1}^{n}(x_i-\bar{x})^2}).$$ Since the variance of \(\hat{\beta}\) is $$\frac{\sigma_\epsilon^2}{n\sigma_x^2}$$ we can use it to calculate the power and sample size for the group sequential design.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


(design1 <- getDesignOneSlope(
  beta = 0.1, n = NA, slope = 0.5,
  stDev = 15, stDevCovariate = 9,
  normalApproximation = FALSE,
  alpha = 0.025))
#>                                                                        
#> Fixed design for one-sample slope                                      
#> Slope under H0: 0, slope under H1: 0.5                                 
#> Standard deviation of residual: 15, standard deviation of covariate: 9 
#> Overall power: 0.9007, overall alpha (1-sided): 0.025                  
#> Drift parameter: 3.273, inflation factor: 1                            
#> Information: 42.84                                                     
#> Number of subjects: 119                                                
#>                                                                        
#>                                 
#> Efficacy boundary (t)     1.980 
#> Efficacy boundary (slope) 0.303 
#> Efficacy boundary (p)     0.0250